It is required to use a program to finish this question Only

It is required to use a program to finish this question. Only programs written in C/C++/Matlab/Octave is acceptable. Hand in the code via CCLE (please put all of them in one file). Print your tabulated results and graphs. Please also print the code out and hand in the hard copy of the code. Consider the following function f: [-1, 1] rightarrow R f(x) = |x| (a) Plot the graph of the function f using MATLAB, and hand in the graph. (b) Given n N\\{0}, define x_n^k = -1 + 2k/n for 0 lessthanorequalto k lessthanorequalto n. Let g_n(x) be the unique polynomial of degree n which results by interpolating the n + 1 data {(x_n^k, f(x_n^k))}0 lessthanorequalto k lessthanorequalto n, i.e. g_n(x_n^k) = f(x_n^k) for all 0 lessthanorequalto k lessthanorequalto n. (You can write a program to do it by any interpolation.) Plot the functions f, g_2, g_3, g_4 and g_5 on the same graph, and hand in the graph. (c) Plot the sequence {g_n(0.3)}1 lessthanorequalto n lessthanorequalto 20, and hand in the graph.

Solution

Ans-

The following theorems were first proposed and proven by Ludwig Sylow in 1872, and published in Mathematische Annalen.

Theorem 1: For every prime factor p with multiplicity n of the order of a finite group G, there exists a Sylow p-subgroup of G, of order pⁿ.

The following weaker version of theorem 1 was first proved by Cauchy, and is known as Cauchy\'s theorem.

Corollary: Given a finite group G and a prime number p dividing the order of G, then there exists an element (and hence a subgroup) of order p in G.^[1]

Theorem 2: Given a finite group G and a prime number p, all Sylow p-subgroups of G are conjugate to each other, i.e. if H and K are Sylow p-subgroups of G, then there exists an element g in G with g¹Hg = K.

Theorem 3: Let p be a prime factor with multiplicity n of the order of a finite group G, so that the order of G can be written as pⁿm, where n > 0 and p does not divide m. Let n_p be the number of Sylow p-subgroups of G. Then the following hold:

Consequences

The Sylow theorems imply that for a prime number p every Sylow p-subgroup is of the same order, pⁿ. Conversely, if a subgroup has order pⁿ, then it is a Sylow p-subgroup, and so is isomorphic to every other Sylow p-subgroup. Due to the maximality condition, if H is any p-subgroup of G, then H is a subgroup of a p-subgroup of order pⁿ.

A very important consequence of Theorem 2 is that the condition n_p = 1 is equivalent to saying that the Sylow p-subgroup of G is a normal subgroup (there are groups that have normal subgroups but no normal Sylow subgroups, such as S₄).

Sylow theorems for infinite groups

There is an analogue of the Sylow theorems for infinite groups. We define a Sylow p-subgroup in an infinite group to be a p-subgroup (that is, every element in it has p-power order) that is maximal for inclusion among all p-subgroups in the group. Such subgroups exist by Zorn\'s lemma.